# Group ring is not isomorphic to 2 by 2 matrices

Let k be an algebraically closed field of characteristic 0. How to prove that $M_2$(k), the k-algebra of 2 by  2 matrices over k, is not isomorphic to the group ring of any finite group G over k.

Any such group must have order 4 and any group of order 4 is abelian so its group ring must be commutative. However, $M_2(k)$ is not commutative.
Indeed, there is always a $k$-algebra morphism $kG\to k$, coming from the trivial representation which, if $kG$ is simple, must be injective: this is only possible if $\dim kG=1$, that is, if $G$ is trivial.